Resolvent expansion for the Schr\"odinger operator on a graph with infinite rays

TL;DR

This paper derives an asymptotic expansion of the resolvent for a Schr"odinger operator on a graph with finite core and infinite rays, clarifying the role of eigenfunctions in threshold behavior.

Contribution

It provides explicit formulas for the resolvent expansion coefficients and links threshold types to eigenfunction growth, simplifying analysis on complex graphs.

Findings

Explicit resolvent expansion coefficients derived

Threshold types classified by eigenfunction growth

Simplified expansion method for graphs without zero eigenvalues

Abstract

We consider the Schr\"odinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$. Precise expressions are obtained for the first few coefficients of the expansion in terms of the generalized eigenfunctions. This result justifies the classification of threshold types solely by growth properties of the generalized eigenfunctions. By choosing an appropriate free operator a priori possessing no zero eigenvalue or zero resonance we can simplify the expansion procedure as much as that on the single discrete half-line.